3.2.67 \(\int \frac {(a+b \log (c x^n))^2 (d+e \log (f x^r))}{x^2} \, dx\) [167]

Optimal. Leaf size=181 \[ -\frac {2 b^2 e n^2 r}{x}-\frac {2 b e n (a+b n) r}{x}-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b^2 e n r \log \left (c x^n\right )}{x}-\frac {2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x} \]

[Out]

-2*b^2*e*n^2*r/x-2*b*e*n*(b*n+a)*r/x-e*(2*b^2*n^2+2*a*b*n+a^2)*r/x-2*b^2*e*n*r*ln(c*x^n)/x-2*b*e*(b*n+a)*r*ln(
c*x^n)/x-b^2*e*r*ln(c*x^n)^2/x-2*b^2*n^2*(d+e*ln(f*x^r))/x-2*b*n*(a+b*ln(c*x^n))*(d+e*ln(f*x^r))/x-(a+b*ln(c*x
^n))^2*(d+e*ln(f*x^r))/x

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2342, 2341, 2413, 14} \begin {gather*} -\frac {e r \left (a^2+2 a b n+2 b^2 n^2\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b e r (a+b n) \log \left (c x^n\right )}{x}-\frac {2 b e n r (a+b n)}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 e n r \log \left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b^2 e n^2 r}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x^2,x]

[Out]

(-2*b^2*e*n^2*r)/x - (2*b*e*n*(a + b*n)*r)/x - (e*(a^2 + 2*a*b*n + 2*b^2*n^2)*r)/x - (2*b^2*e*n*r*Log[c*x^n])/
x - (2*b*e*(a + b*n)*r*Log[c*x^n])/x - (b^2*e*r*Log[c*x^n]^2)/x - (2*b^2*n^2*(d + e*Log[f*x^r]))/x - (2*b*n*(a
+ b*Log[c*x^n])*(d + e*Log[f*x^r]))/x - ((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x^2} \, dx &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-(e r) \int \frac {-a^2 \left (1+\frac {2 b n (a+b n)}{a^2}\right )-2 b (a+b n) \log \left (c x^n\right )-b^2 \log ^2\left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}-(e r) \int \left (\frac {-a^2-2 a b n-2 b^2 n^2}{x^2}-\frac {2 b (a+b n) \log \left (c x^n\right )}{x^2}-\frac {b^2 \log ^2\left (c x^n\right )}{x^2}\right ) \, dx\\ &=-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}+\left (b^2 e r\right ) \int \frac {\log ^2\left (c x^n\right )}{x^2} \, dx+(2 b e (a+b n) r) \int \frac {\log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b e n (a+b n) r}{x}-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}+\left (2 b^2 e n r\right ) \int \frac {\log \left (c x^n\right )}{x^2} \, dx\\ &=-\frac {2 b^2 e n^2 r}{x}-\frac {2 b e n (a+b n) r}{x}-\frac {e \left (a^2+2 a b n+2 b^2 n^2\right ) r}{x}-\frac {2 b^2 e n r \log \left (c x^n\right )}{x}-\frac {2 b e (a+b n) r \log \left (c x^n\right )}{x}-\frac {b^2 e r \log ^2\left (c x^n\right )}{x}-\frac {2 b^2 n^2 \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \left (d+e \log \left (f x^r\right )\right )}{x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.08, size = 138, normalized size = 0.76 \begin {gather*} -\frac {a^2 d+2 a b d n+2 b^2 d n^2+a^2 e r+4 a b e n r+6 b^2 e n^2 r+e \left (a^2+2 a b n+2 b^2 n^2\right ) \log \left (f x^r\right )+b^2 \log ^2\left (c x^n\right ) \left (d+e r+e \log \left (f x^r\right )\right )+2 b \log \left (c x^n\right ) \left (a (d+e r)+b n (d+2 e r)+e (a+b n) \log \left (f x^r\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])^2*(d + e*Log[f*x^r]))/x^2,x]

[Out]

-((a^2*d + 2*a*b*d*n + 2*b^2*d*n^2 + a^2*e*r + 4*a*b*e*n*r + 6*b^2*e*n^2*r + e*(a^2 + 2*a*b*n + 2*b^2*n^2)*Log
[f*x^r] + b^2*Log[c*x^n]^2*(d + e*r + e*Log[f*x^r]) + 2*b*Log[c*x^n]*(a*(d + e*r) + b*n*(d + 2*e*r) + e*(a + b
*n)*Log[f*x^r]))/x)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.47, size = 8407, normalized size = 46.45

method	result	size
risch	\(\text {Expression too large to display}\)	\(8407\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))^2*(d+e*ln(f*x^r))/x^2,x,method=_RETURNVERBOSE)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 227, normalized size = 1.25 \begin {gather*} -b^{2} {\left (\frac {r}{x} + \frac {\log \left (f x^{r}\right )}{x}\right )} e \log \left (c x^{n}\right )^{2} - 2 \, a b {\left (\frac {r}{x} + \frac {\log \left (f x^{r}\right )}{x}\right )} e \log \left (c x^{n}\right ) - 2 \, b^{2} d {\left (\frac {n^{2}}{x} + \frac {n \log \left (c x^{n}\right )}{x}\right )} - 2 \, {\left (\frac {{\left (r \log \left (x\right ) + 3 \, r + \log \left (f\right )\right )} n^{2}}{x} + \frac {n {\left (2 \, r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} \log \left (c x^{n}\right )}{x}\right )} b^{2} e - \frac {2 \, a b n {\left (2 \, r + \log \left (f\right ) + \log \left (x^{r}\right )\right )} e}{x} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{x} - \frac {2 \, a b d n}{x} - \frac {a^{2} r e}{x} - \frac {2 \, a b d \log \left (c x^{n}\right )}{x} - \frac {a^{2} e \log \left (f x^{r}\right )}{x} - \frac {a^{2} d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*(d+e*log(f*x^r))/x^2,x, algorithm="maxima")

[Out]

-b^2*(r/x + log(f*x^r)/x)*e*log(c*x^n)^2 - 2*a*b*(r/x + log(f*x^r)/x)*e*log(c*x^n) - 2*b^2*d*(n^2/x + n*log(c*
x^n)/x) - 2*((r*log(x) + 3*r + log(f))*n^2/x + n*(2*r + log(f) + log(x^r))*log(c*x^n)/x)*b^2*e - 2*a*b*n*(2*r
+ log(f) + log(x^r))*e/x - b^2*d*log(c*x^n)^2/x - 2*a*b*d*n/x - a^2*r*e/x - 2*a*b*d*log(c*x^n)/x - a^2*e*log(f
*x^r)/x - a^2*d/x

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 316, normalized size = 1.75 \begin {gather*} -\frac {b^{2} n^{2} r e \log \left (x\right )^{3} + 2 \, b^{2} d n^{2} + 2 \, a b d n + a^{2} d + {\left (6 \, b^{2} n^{2} + 4 \, a b n + a^{2}\right )} r e + {\left (b^{2} r e + b^{2} d\right )} \log \left (c\right )^{2} + {\left (2 \, b^{2} n r e \log \left (c\right ) + b^{2} n^{2} e \log \left (f\right ) + b^{2} d n^{2} + {\left (3 \, b^{2} n^{2} + 2 \, a b n\right )} r e\right )} \log \left (x\right )^{2} + 2 \, {\left (b^{2} d n + a b d + {\left (2 \, b^{2} n + a b\right )} r e\right )} \log \left (c\right ) + {\left (b^{2} e \log \left (c\right )^{2} + 2 \, {\left (b^{2} n + a b\right )} e \log \left (c\right ) + {\left (2 \, b^{2} n^{2} + 2 \, a b n + a^{2}\right )} e\right )} \log \left (f\right ) + {\left (b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} d n^{2} + 2 \, a b d n + {\left (6 \, b^{2} n^{2} + 4 \, a b n + a^{2}\right )} r e + 2 \, {\left (b^{2} d n + {\left (2 \, b^{2} n + a b\right )} r e\right )} \log \left (c\right ) + 2 \, {\left (b^{2} n e \log \left (c\right ) + {\left (b^{2} n^{2} + a b n\right )} e\right )} \log \left (f\right )\right )} \log \left (x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*(d+e*log(f*x^r))/x^2,x, algorithm="fricas")

[Out]

-(b^2*n^2*r*e*log(x)^3 + 2*b^2*d*n^2 + 2*a*b*d*n + a^2*d + (6*b^2*n^2 + 4*a*b*n + a^2)*r*e + (b^2*r*e + b^2*d)
*log(c)^2 + (2*b^2*n*r*e*log(c) + b^2*n^2*e*log(f) + b^2*d*n^2 + (3*b^2*n^2 + 2*a*b*n)*r*e)*log(x)^2 + 2*(b^2*
d*n + a*b*d + (2*b^2*n + a*b)*r*e)*log(c) + (b^2*e*log(c)^2 + 2*(b^2*n + a*b)*e*log(c) + (2*b^2*n^2 + 2*a*b*n
+ a^2)*e)*log(f) + (b^2*r*e*log(c)^2 + 2*b^2*d*n^2 + 2*a*b*d*n + (6*b^2*n^2 + 4*a*b*n + a^2)*r*e + 2*(b^2*d*n
+ (2*b^2*n + a*b)*r*e)*log(c) + 2*(b^2*n*e*log(c) + (b^2*n^2 + a*b*n)*e)*log(f))*log(x))/x

________________________________________________________________________________________

Sympy [A]
time = 1.21, size = 280, normalized size = 1.55 \begin {gather*} - \frac {a^{2} d}{x} - \frac {a^{2} e r}{x} - \frac {a^{2} e \log {\left (f x^{r} \right )}}{x} - \frac {2 a b d n}{x} - \frac {2 a b d \log {\left (c x^{n} \right )}}{x} - \frac {4 a b e n r}{x} - \frac {2 a b e n \log {\left (f x^{r} \right )}}{x} - \frac {2 a b e r \log {\left (c x^{n} \right )}}{x} - \frac {2 a b e \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{x} - \frac {2 b^{2} d n^{2}}{x} - \frac {2 b^{2} d n \log {\left (c x^{n} \right )}}{x} - \frac {b^{2} d \log {\left (c x^{n} \right )}^{2}}{x} - \frac {6 b^{2} e n^{2} r}{x} - \frac {2 b^{2} e n^{2} \log {\left (f x^{r} \right )}}{x} - \frac {4 b^{2} e n r \log {\left (c x^{n} \right )}}{x} - \frac {2 b^{2} e n \log {\left (c x^{n} \right )} \log {\left (f x^{r} \right )}}{x} - \frac {b^{2} e r \log {\left (c x^{n} \right )}^{2}}{x} - \frac {b^{2} e \log {\left (c x^{n} \right )}^{2} \log {\left (f x^{r} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))**2*(d+e*ln(f*x**r))/x**2,x)

[Out]

-a**2*d/x - a**2*e*r/x - a**2*e*log(f*x**r)/x - 2*a*b*d*n/x - 2*a*b*d*log(c*x**n)/x - 4*a*b*e*n*r/x - 2*a*b*e*
n*log(f*x**r)/x - 2*a*b*e*r*log(c*x**n)/x - 2*a*b*e*log(c*x**n)*log(f*x**r)/x - 2*b**2*d*n**2/x - 2*b**2*d*n*l
og(c*x**n)/x - b**2*d*log(c*x**n)**2/x - 6*b**2*e*n**2*r/x - 2*b**2*e*n**2*log(f*x**r)/x - 4*b**2*e*n*r*log(c*
x**n)/x - 2*b**2*e*n*log(c*x**n)*log(f*x**r)/x - b**2*e*r*log(c*x**n)**2/x - b**2*e*log(c*x**n)**2*log(f*x**r)
/x

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (190) = 380\).
time = 6.80, size = 392, normalized size = 2.17 \begin {gather*} -\frac {b^{2} n^{2} r e \log \left (x\right )^{3} + 3 \, b^{2} n^{2} r e \log \left (x\right )^{2} + 2 \, b^{2} n r e \log \left (c\right ) \log \left (x\right )^{2} + b^{2} n^{2} e \log \left (f\right ) \log \left (x\right )^{2} + 6 \, b^{2} n^{2} r e \log \left (x\right ) + 4 \, b^{2} n r e \log \left (c\right ) \log \left (x\right ) + b^{2} r e \log \left (c\right )^{2} \log \left (x\right ) + 2 \, b^{2} n^{2} e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) \log \left (x\right ) + b^{2} d n^{2} \log \left (x\right )^{2} + 2 \, a b n r e \log \left (x\right )^{2} + 6 \, b^{2} n^{2} r e + 4 \, b^{2} n r e \log \left (c\right ) + b^{2} r e \log \left (c\right )^{2} + 2 \, b^{2} n^{2} e \log \left (f\right ) + 2 \, b^{2} n e \log \left (c\right ) \log \left (f\right ) + b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 2 \, b^{2} d n^{2} \log \left (x\right ) + 4 \, a b n r e \log \left (x\right ) + 2 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 2 \, a b r e \log \left (c\right ) \log \left (x\right ) + 2 \, a b n e \log \left (f\right ) \log \left (x\right ) + 2 \, b^{2} d n^{2} + 4 \, a b n r e + 2 \, b^{2} d n \log \left (c\right ) + 2 \, a b r e \log \left (c\right ) + b^{2} d \log \left (c\right )^{2} + 2 \, a b n e \log \left (f\right ) + 2 \, a b e \log \left (c\right ) \log \left (f\right ) + 2 \, a b d n \log \left (x\right ) + a^{2} r e \log \left (x\right ) + 2 \, a b d n + a^{2} r e + 2 \, a b d \log \left (c\right ) + a^{2} e \log \left (f\right ) + a^{2} d}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))^2*(d+e*log(f*x^r))/x^2,x, algorithm="giac")

[Out]

-(b^2*n^2*r*e*log(x)^3 + 3*b^2*n^2*r*e*log(x)^2 + 2*b^2*n*r*e*log(c)*log(x)^2 + b^2*n^2*e*log(f)*log(x)^2 + 6*
b^2*n^2*r*e*log(x) + 4*b^2*n*r*e*log(c)*log(x) + b^2*r*e*log(c)^2*log(x) + 2*b^2*n^2*e*log(f)*log(x) + 2*b^2*n
*e*log(c)*log(f)*log(x) + b^2*d*n^2*log(x)^2 + 2*a*b*n*r*e*log(x)^2 + 6*b^2*n^2*r*e + 4*b^2*n*r*e*log(c) + b^2
*r*e*log(c)^2 + 2*b^2*n^2*e*log(f) + 2*b^2*n*e*log(c)*log(f) + b^2*e*log(c)^2*log(f) + 2*b^2*d*n^2*log(x) + 4*
a*b*n*r*e*log(x) + 2*b^2*d*n*log(c)*log(x) + 2*a*b*r*e*log(c)*log(x) + 2*a*b*n*e*log(f)*log(x) + 2*b^2*d*n^2 +
4*a*b*n*r*e + 2*b^2*d*n*log(c) + 2*a*b*r*e*log(c) + b^2*d*log(c)^2 + 2*a*b*n*e*log(f) + 2*a*b*e*log(c)*log(f)
+ 2*a*b*d*n*log(x) + a^2*r*e*log(x) + 2*a*b*d*n + a^2*r*e + 2*a*b*d*log(c) + a^2*e*log(f) + a^2*d)/x

________________________________________________________________________________________

Mupad [B]
time = 4.01, size = 181, normalized size = 1.00 \begin {gather*} -\ln \left (f\,x^r\right )\,\left (\ln \left (c\,x^n\right )\,\left (\frac {2\,a\,b\,e}{x}+\frac {2\,b^2\,e\,n}{x}\right )+\frac {a^2\,e}{x}+\frac {2\,b^2\,e\,n^2}{x}+\frac {b^2\,e\,{\ln \left (c\,x^n\right )}^2}{x}+\frac {2\,a\,b\,e\,n}{x}\right )-\frac {a^2\,d+2\,b^2\,d\,n^2+a^2\,e\,r+6\,b^2\,e\,n^2\,r+2\,a\,b\,d\,n+4\,a\,b\,e\,n\,r}{x}-\frac {2\,b\,\ln \left (c\,x^n\right )\,\left (a\,d+b\,d\,n+a\,e\,r+2\,b\,e\,n\,r\right )}{x}-\frac {b^2\,{\ln \left (c\,x^n\right )}^2\,\left (d+e\,r\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*log(f*x^r))*(a + b*log(c*x^n))^2)/x^2,x)

[Out]

- log(f*x^r)*(log(c*x^n)*((2*a*b*e)/x + (2*b^2*e*n)/x) + (a^2*e)/x + (2*b^2*e*n^2)/x + (b^2*e*log(c*x^n)^2)/x
+ (2*a*b*e*n)/x) - (a^2*d + 2*b^2*d*n^2 + a^2*e*r + 6*b^2*e*n^2*r + 2*a*b*d*n + 4*a*b*e*n*r)/x - (2*b*log(c*x^
n)*(a*d + b*d*n + a*e*r + 2*b*e*n*r))/x - (b^2*log(c*x^n)^2*(d + e*r))/x

________________________________________________________________________________________